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Let (K α: αω) be a system of varieties definable by schemas. We characterize the amalgamation base, strong amalgamation base, and super amalgamation base of the class \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $S\mathfrak{N}\mathfrak{r}_\alpha $ \end{document} K α+ω at this abstract level.

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We construct an infinite dimensional quasi-polyadic equality algebra \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak{A}$ \end{document} such that its cylindric reduct is representable, while \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak{A}$ \end{document} itself is not representable.

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Let α be an infinite ordinal. Let RCAα denote the variety of representable cylindric algebras of dimension α. Modifying Andréka’s methods of splitting, we show that the variety RQEAα of representable quasi-polyadic equality algebras of dimension α is not axiomatized by a set of universal formulas containing only finitely many variables over the variety RQAα of representable quasi-polyadic algebras of dimension α. This strengthens a seminal result due to Sain and Thompson, answers a question posed by Andréka, and lifts to the transfinite a result of hers proved for finite dimensions > 2. Using the modified method of splitting, we show that all known complexity results on universal axiomatizations of RCAα (proved by Andréka) transfer to universal axiomatizations of RQEAα. From such results it can be inferred that any algebraizable extension of Lω,ω is severely incomplete if we insist on Tarskian square semantics. Ways of circumventing the strong non-negative axiomatizability results hitherto obtained in the first part of the paper, such as guarding semantics, and /or expanding the signature of RQEAω by substitutions indexed by transformations coming from a finitely presented subsemigroup of (ωω, ○) containing all transpositions and replacements, are surveyed, discussed, and elaborated upon.

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Abstract

Fix 2 < n < ω and let CAn denote the class of cyindric algebras of dimension n. Roughly CAn is the algebraic counterpart of the proof theory of first order logic restricted to the first n variables which we denote by Ln. The variety RCAn of representable CAns reflects algebraically the semantics of L n. Members of RCAn are concrete algebras consisting of genuine n-ary relations, with set theoretic operations induced by the nature of relations, such as projections referred to as cylindrifications. Although CAn has a finite equational axiomatization, RCAn is not finitely axiomatizable, and it generally exhibits wild, often unpredictable and unruly behavior. This makes the theory of CAn substantially richer than that of Boolean algebras, just as much as Lω,ω is richer than propositional logic. We show using a so-called blow up and blur construction that several varieties (in fact infinitely many) containing and including the variety RCAn are not atom-canonical. A variety V of Boolean algebras with operators is atom canonical, if whenever 𝔄 V is atomic, then its Dedekind-MacNeille completion, sometimes referred to as its minimal completion, is also in V. From our hitherto obtained algebraic results we show, employing the powerful machinery of algebraic logic, that the celebrated Henkin-Orey omitting types theorem, which is one of the classical first (historically) cornerstones of model theory of L ω,ω, fails dramatically for L n even if we allow certain generalized models that are only locallly clasfsical. It is also shown that any class K such that NrnCAωCRCAn¯K¯ScNrnCAn+3 , where CRCAn is the class of completely representable CAns, and Sc denotes the operation of forming dense (complete) subalgebras, is not elementary. Finally, we show that any class K such that SdRaCAω¯K¯ScRaCA5 is not elementary, where Sd denotes the operation of forming dense subalgebra.

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